I I'want to find a webcam for my HP desktop computer that finds a good clear picture on it? But I don't want to spend more than $50?

Hello there, and great question.  When you want to webcam it's important to find a webcam that has  good quality so that the image isn't all pixelated when you're doing a video chat.  Like a normal digital camera the more megapixels the webcam has the clearer the quality will be.  Logitech is a good brand for webcams, they are reliable,  everyone I know that has a webcam it's by Logitech.  Some of the webcams can get a bit pricey but there are webcams within your price range with good quality. I suggest you check out these webcams: 

Logitech Webcam C250
"Grab the Logitech Webcam C250 that features crisp, colorful video with 1.3 megapixel sensor. Your loved ones will see you in crisp, colorful video when you talk to them, or you can take photos to share later. The Logitech Vid is free, fast and easy for you and anyone to call. It sets up when you set up the webcam. You can be heard loud and clear thanks to a built-in micrphone that reduces background noise."

QuickCam Pro
QuickCam Pro from Logitech is a webcam designed for use with your notebook. It uses AutoFocus with Carl Zeiss optics to capture every detail. Its RightLight 2 technology ensures great looking images, even in poor lighting where as RightSound technology with echo-cancellation and noise reduction makes conversations crystal clear and natural. You can personalize video calls with Logitech Video Effects filters, avatars and face accessories that mirror your expressions and motion. Additionally, an ultra-wide field of view and automatic face tracking always keeps you right in the middle of the action. The QuickCam Pro comes with a desktop stand to bring your webcam up to eye level and a travel case to protect your camera when not in use.